Long summation question, including sets

Question

I have a really long question I'm absolutely stuck on, I don't even know where to begin:

Given:

$n \in \mathbb{Z}, \geq 2$

let $S$ be the set of all nonempty subsets of {2,3,...,n}. For each $S_i \in S$ let $P_i$ be thje product of the elements of $S_i$ prove or disprove:

$\sum\limits_{i=1}^{2^{n-1}-1} P_i = \frac{(n+1)!}{2} -1$

Answer 1

For a set $S$, consider the product $\prod_{x \in S}(1+x)$. Each term in this expansion is the product of elements of one proper subset of $S$, except for the term $1$. Applying this to your set and simplifying gives the answer.

Answer 2

HINT: You can prove it quite straightforwardly by induction on $n$ if you notice that the non-empty subsets of $\{2,\ldots,n,n+1\}$ are of three kinds:

• non-empty subsets of $\{2,\ldots,n\}$;
• $S\cup\{n+1\}$, where $S$ is a non-empty subset of $\{2,\ldots,n\}$; and
• $\{n+1\}$.